and such that . Thus, (1.1) is satisfied, for all . In particular, if either and , or and , then the inequality (1.1) holds. Thus, is equiregulated.
The next result describes an interesting property of equiregulated sets of . Such result can be found in [97, Proposition 3.8].
Theorem 1.12:Assume that a set is equiregulated and, for any , there is a number such that
Then, there is a constant such that, for every ,
Proof. Take as the set of all numbers fulfilling the condition that there exists a positive number for which we have , for every and every Since is an equiregulated set, there exists such that for every and every . From this fact and the hypotheses, we can infer that for every and every , we have
Then, .
Let . The equiregulatedness of implies that there exists such that for every and . Take . Thus, for every ,
